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220 \label{axiom:morphisms} |
220 \label{axiom:morphisms} |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
222 the category of $k$-balls and |
222 the category of $k$-balls and |
223 homeomorphisms to the category of sets and bijections. |
223 homeomorphisms to the category of sets and bijections. |
224 \end{axiom} |
224 \end{axiom} |
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225 |
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226 Note that the functoriality in the above axiom allows us to operate via |
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227 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
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228 The action of these homeomorphisms gives the ``strong duality" structure. |
225 |
229 |
226 Next we consider domains and ranges of $k$-morphisms. |
230 Next we consider domains and ranges of $k$-morphisms. |
227 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
231 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
228 into domain and range --- the duality operations can convert domain to range and vice-versa. |
232 into domain and range --- the duality operations can convert domain to range and vice-versa. |
229 Instead, we will use a unified domain/range, which we will call a ``boundary". |
233 Instead, we will use a unified domain/range, which we will call a ``boundary". |